OnC-det spectral andC-det-convex matrices
Natália Bebiano, Yiu‐Tung Poon, João da Providência · Linear and Multilinear Algebra · 1988
Denote by the group of the n × n unitary matrices. Let A be a complex n × n matrix with eigenvalues α1,⋯αn and C = diag(γ1,⋯γn)c = (γ1,⋯γn)ε. The set of complex numbers will be called the determinantal range of A Let P c (A) be the convex hull of , where Sn is the symmetric group of degree n. In this note, the concepts of c-determinantal radius (d c (A)) and of c-det-spectral radius are introduced. The matrices for which hold are investigated. Some new analogies between Δ c (A) and the well-known c-numerical range of A are found. Also, the interesting interplay between the geometric properties of Δ c (A) and the algebraic properties of A is re-examined and strengthened by the results obtained here. A necessary and sufficient condition for the convexity of Δ c (A) in terms of the eigenvalues of A and C for 3 × 3 normal matrices is given. A sufficient condition for the convexity of Δc(A) when A and C are, respectively, a 3 × 3 matrix and a 3 × 3 normal matrix is presented.